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Three Random Numbers

Q: You play a game with a friend where he chooses two random numbers between 0 and 1. Next you choose a random number between 0 and 1. If your number falls between the prior two numbers you win. What is the probability that you would win?

A: Consider the number line between 0 and 1 shown in figure below

Let \(x\) and \(y\) be the two numbers chosen. The probability of a win, i.e. choosing a number in the range between the two chosen numbers can be estimated as \(\frac{|y-x|}{1}\). Note, we have chosen the modulus operation because \(x\) could be greater than \(y\) or vice versa. The feasible region of numbers to be chosen to win, is the ratio of the absolute difference between \(x\) and \(y\) divided by the total possible range, which is 1. In order to estimate the probability that a third chosen number will lie between the two we integrate out (a double integral) between the ranges of \([0,1]\). This is estimated as
$$
P(\text{win}) = \int_{0}^{1}\int_{0}^{1}|y - x| dx dy
$$
To evaluate the above integral, we split the inner integral as follows
$$
P(\text{win}) = \int_{0}^{1}\Big[\int_{0}^{y}(y - x)dx + \int_{y}^{1}(x-y)dx\Big]dy
$$
The inner integral evaluates to \(y^2 - y +\frac{1}{2}\) which on further integration between the ranges of \([0,1]\) yields
$$
P(\text{win}) = [\frac{y^3}{3} - \frac{y^2}{2} + \frac{y}{2}]_{0}^{1} = \frac{1}{3}
$$
which is the sought probability.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Comments

An alternative solution: the problem can be rephrased as "Draw three points in [0,1] at random. What is the probability of the third point falling between the first two?". Ordering the chosen points by magnitude results in one of 3!=6 possible permutations. By symmetry, each permutation is equally likely to occur. The event we are interested in is comprised of two of the permutations ("1-3-2" and "2-3-1"). Therefore the probability of this event is 2/6 = 1/3.

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.